Pipeline Design

On a New Methodology for the Design

and Operations Planning of Natural

Gas Pipeline Networks

Débora Campos de Faria, Kristy Booth, Chase Waite and

Miguel Bagajewicz*

Department of Chemical Biological and Materials Engineering

University of Oklahoma

Kewords: Pipeline Design

(*) Corresponding author: [email protected]

mailto:[email protected]

ABSTRACT

This paper presents a new methodology to design natural gas pipelines over a long time

horizon considering determining timely expansions by the addition of compressors. The

problem is framed in the context of a capacity expansion problem and is solved using a

non-linear mixed integer model. We compare our methodology to the current state-of-

the-art methodologies, highlighting the limitations of the latter and showing the

advantages of the former, especially in the case of branched networks.

1. INTRODUCTION

The economical design of pipelines is important because the cost for

transmission and distribution is almost half of the total price the end user pays for

natural gas. The United States consumes between 1.5 and 2.5 million MMcf of natural

gas per month depending on the season, and this huge volume has to be transported

somehow to the end user1. About 97 % of the natural gas consumed in the U.S. is

produced at various places in North America. So, there is no hub where all the gas is

produced, the producing sites are constantly changing their rates and producing status

and new wells are also constantly being discovered. Currently, it is common to

examine the feasibility of projects based on J-curves which analyze the Total Annual

Cost per MCF transported versus the flow rate. This type of analysis can be sufficient

for a simple project with no ramifications and a single pipe. But when analyzing a more

complicated network, J-curves become exponentially more time consuming, and many

times will not even give the correct optimum.

2. PROBLEM STATEMENT

We now discuss different cases:

a. Conventional structure

In this case the pipeline routes and compressors locations are predefined, but

initial capacities and expansion capacities (size and timing) are not. Compressors are

located only at the supplier points and or some consumer or recompression station

points. We want to know the optimum pipeline diameters for each section and the

compressors capacities at the suppliers. Branching and non branching structures can

be analyzed. The following diagram serves as an example of the conventional

design approach, from the planning stage to some time t*.

t=t*

F=45

F=75 F=57

F=32

D=28”

D=32” D = 32” Cap?

W?

D?

D? D?

F=80 F=63

t=t*

Consumer point without a compressor

Compressor Station

Cap = Compressor size

D = Pipe diameter

F = Flow rate

W = Required work

D = 32” D = 32”

D = 28”

t=0

Input Data

Cap = 4000

W = 2500

Cap = 4000

W = 3500

In the above example, the maximum compressor capacity and required diameters are

computed. When the demand increases at time t*, the work of the compressor

increases, while the capacity and diameters remain constant.

b. Compressor location

The pipeline routes are predefined. We want to know the pipeline sections

diameters and where, when and with which capacity a compressor should be

installed.

t=t*

F=45

F=75 F=57 F=32

D=28”

D=32” D = 32” Cap?

W?

D?

D? D?

F=80 F=63

t=t*

t=0

Consumer point with possible compressor

These locations are predefined as consumer

points but may require compressors under

future demands

W1 = Work required at first compressor

Cap1 = First compressor size

W2 = Work required at second compressor

Cap2 = Second compressor size

F = Flow rate, MMSCFD

D = 32” D = 32”

D = 28”

Cap = 4000

W = 2500

Input Data

Cap1 = 4000 Cap2 =2000

W1 = 3500 W2 =1500

During the design phase, the consumer points are predefined. It is unknown if a

compressor is required at any of these points. By time t=0, a compressor is required at

the supply point, yet the possible increase in demand may require further addition of

compressors at the consumer points. At time t=t*, a second compressor is added, and

all other consumer points are no longer available for adding compressors.

c. Loop and compressors locations

The pipeline routes are predefined. We want to know the pipeline diameters; where,

when and with which capacity a compressor should be installed; and, where, when and

which capacity loops should be added. During the design phase, two possible loops are

considered in this example. The route has already been determined, but an increase in

demand may require one or both of the loops at a later time. At time t=0, adding the

supply compressor determines that only one loop will be further considered. The

remaining consumer points are still available for adding compressor as required. At

time t=t* the increase in demand requires greater work at the compressor. The pipe

diameters at t=t* are slightly varied from those of t=0. The maximum required diameter

must be chosen for operation.

t=t*

F=45

F=75 F=57 F=32

D=28”

D=32” D = 32” Cap?

W?

D?

D? D?

F=80 F=63

t=t*

t=0




future demands

Possible loop






D = 32” D = 32”

D = 24”

Cap = 4000

W = 2500

Input Data

Cap1 = 4000 Cap2 =2000

W1 = 3500 W2 =1500

D = 28”

d. Alternative routes, loops and compressors locations

Only the consumers and suppliers distances are predefined. We want to know

where, when and with what capacity pipelines (including loops) and compressors

should be installed. The possible loops and alternative routes are shown in the input

data. At time t=0, an alternative route is chosen, but the consumer points are shown

as having the option of adding a compressor. By t=t*, a second compressor is added

and the other consumer points are not available for compressor addition.

t=t*

F=45

F=75 F=57 F=32

D=28”

D=32” D = 32” Cap?

W?

D?

D? D?

F=80 F=63

t=t*

t=0




future demands

Possible loop or alternative route






D = 32” D = 32”

D = 28”

Cap = 4000

W = 2500

Input Data

Cap1 = 4000 Cap2 =2000

W1 = 3500 W2 =1500

D = 24”

In this paper only cases a and b will be approached. The others are subject for

further study.

3. PIPELINE DESIGN BASICS

Typically, to calculate the pressure drops, the literature recommends using the

Panhandle A equation (for Reynolds numbers between 5 and 11 million)

(1)

and the Panhandle B equation (suitable for Reynolds numbers between 4 and 40

million)4.

(2)

In these equations,

Q volume flow rate (SCFD)

E the Pipeline efficiency

Pb the base pressure (psia)

Tb the base temperature (°R)

P1 the upstream pressure (psia)

P2 the downstream pressure (psia)

G the gas gravity

Tf the average gas flow temperature (°R)

Le the equivalent length of pipe segment (miles)

Z the gas compressibility factor (dimensionless)

D the pipe inside diameter (in)

e = base of natural logarithms (e=2.718…)

s = elevation adjustment parameter, dimensionless

The elevation adjustment parameter is found from the following equation:

Where

H1 is the upstream elevation (m)

H2 is the downstream elevation (m)

The panhandle equations were derived from simplifications of the Bernoulli equation.

In its most inclusive form, the Bernoulli is as follows:

In order to arrive at the panhandle equations above, specific transmission factors are

used. For the Panhandle A equation, the transmission factor is 6.872(Re)0.0730

. For the

Panhandle B equation, the transmission factor is 16.49(Re)0.01961

. The constant values

of the panhandle equations are dependent on an assumed viscosity, as well as the

assumption that .

Qb is assumed to be at base conditions, with a base temperature of 520ºR and a base

pressure of 14.7 psia.

Besides these pressure drop correlations, simulation software can also be used. Here

we compare the results using the Panhandle correlations to the ones found using

simulations from Pro/II. The simulations used the Beggs-Brill-Moody pressure drop

correlation, which is good for mostly vapor hydrocarbon systems with small elevation

changes, very similar to this situation. We used the Soave-Redlich-Kwong

thermodynamic equation of state, which is suitable for hydrocarbon systems such as the

one being examined. A negligible pipe roughness was assumed, along with a length of

120 miles, a composition listed in Table 1, an outlet pressure of 800 psig, an inlet

temperature of 60° F, a viscosity of 7.53 x 10-7

lb/ft/s, and a pipeline efficiency of

0.922,3,4

. An overall coefficient of heat transfer was assumed to be 0.3 Btu/hr-ft2-°F

based on rough calculations using the correlation by Churchill and Chu6.

Table 1

Natural Gas Composition Used5

Natural Gas Component Mole Fraction

C1 0.949

C2 0.025

C3 .002

N2 0.016

CO2 0.007

C4 0.0003

iC4 0.0003

C5 0.0001

iC5 0.0001

O2 0.0002

The two independent methods, Pro-II simulation and analytical correlations,

were used to predict these pressure drops. Both methods were analyzed by the same

parameters: flow rate, pipe length, pipe diameter, and downstream pressure. The two

methods supplied an estimate of the outlet compressor pressure, which produced two

distinct pressure drops for each comparison. The comparison was then made for

different pipe diameters and flow rates.

Figure 1: Comparison of pressure drops found using Panhandle equations and Pro/II –

a) No elevation. b) Considering elevation.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

125 175 225 275 325 375 425

Pre

ssure

Dro

p (

psi

a)

Flow Rate (MMSCFD)

A) Pressure Drop Without Elevation Change

NPS = 16; Pro-II NPS = 16; Analytical

NPS =18; Pro-II NPS = 18; Analytical



Figure 2: Comparison of pressure errors with respect to Pro-II using a pipeline

efficiency of 0.92 – a) No elevation. b) Considering elevation.

0

500

1000

1500

2000

125 175 225 275 325 375 425

Pre

ssure

Dro

p (

psi

a)

Flow Rate (MMSCFD)

B) Pressure Drop 0.15 km Elevation Change


NPS =18; Pro-II NPS = 18; Analytical



0

5

10

15

20

25

30

35

40

45

50

150 200 250 300 350 400

Per

cen

t E

rro

r

Flowrate MMSCFD

A) Pressure Drop ErrorNo Elevation Change

NPS = 16

NPS =18

NPS =20

NPS = 22

0

5

10

15

20

25

30

150 200 250 300 350 400

Per

cen

t E

rro

r

Flowrate MMSCFD

B) Pressure Drop Error 0.150 km Elevation Change

NPS = 16

NPS = 18

NPS = 20

NPS =22

In order to analyze the effects of pipeline efficiency on the accuracy of the

Panhandle equations, the pipeline efficiency was changed using the solver function in

Microsoft Excel to minimize the sum of the errors graphed below in Figure 3.

Figure 3: Comparison of pressure errors with respect to Pro-II using an adjusted

pipeline efficiency – a) No elevation. b) Considering elevation.

0

5

10

15

20

25

30

35

40

45

50

150 200 250 300 350 400

Per

cen

t E

rro

r

Flowrate MMSCFD

A) Pressure Drop Error, E = 0.972 No Elevation Change

NPS = 16

NPS =18

NPS =20

NPS = 22

0

5

10

15

20

25

30

150 200 250 300 350 400

Per

cen

t E

rro

r

Flowrate MMSCFD

B) Pressure Drop Error, E = 0.964 0.150 km Elevation Change

NPS = 16

NPS = 18

NPS = 20

NPS =22

By using solver it was possible to minimize the pressure drop error to around

2% or less; however, this still requires the use of a simulator to find a more accurate

Panhandle equation. It seems redundant to use a simulator to adjust a correlation that

should be able to provide the pressure drop without the use of any simulator. Therefore,

for our J-curves we use Pro-II to determine the pressure drop across pipelines.

If the Panhandle equation was used, then the following equation can be used to

calculate the required horsepower of the compressor

(3)

Where

γ is the ratio of specific heats of gas, dimensionless

T1 is the suction temperature of gas (°R)

P1 is the suction pressure of gas (psia)

P2 is the discharge pressure of gas (psia)

Z1 is the compressibility of gas at suction conditions

Z2 is compressibility of gas at discharge conditions

na is the compressor adiabatic efficiency4

Typically, the compressors used are centrifugal which operate at a maximum of

80% efficiency.

4. CONVENTIONAL DESIGN APPROACH

In this section we review the existing recommended procedure for optimizing a

pipeline network. Traditionally, pressure drop correlations are used to determine the

work required by the compressor. The usual situation described in the literature is that

of a single pipe with no branches and one compression station at the beginning. Costs

are then directly estimated for each diameter and translated into what is known as a J-

Curve, which is a graph of the Total Annual Cost ($ per Mcf delivered) versus flow rate

(Figure 1). The range of flow rates can be a range of expected flows due to increased

future demands, stationary changes, or even day vs. night fluctuations. To obtain the

total cost for each flow rate, capital and operating costs are used. The capital costs are

computed by annualizing the compressor as well as the pipe fixed capital investment.

Operating costs are related to the power consumed, as well as maintenance and

overhead costs, such as any materials required during daily operation or labor expenses

not incurred during installation. Interstage cooling costs are included as an operating

expense, if required, as well as routine maintenance to compressor stations or pipeline.

Figure 4: A ‘J-Curve’ showing the total pipeline cost per Mcf versus flow rate for

four different pipe sizes

The pipe material cost is given by4:

where:

D is the diameter (in)

T is the wall thickness (in)

L is the pipe length (mi)

$0.59

$0.58

$0.00

$1.00

$2.00

$3.00

$4.00

$5.00

50 100 150 200 250 300 350 400 450 500

To

tal A

nnual

Co

st p

er M

CF

Flow Rate (MMSCFD)

J-Curve

NPS = 18

NPS = 16

NPS = 20

NPS = 22

C is the material cost ($/ton)

In turn, the pipe installation cost is found using the values given in Table 24:

Typical Pipe Installation Costs

NPS Cost, $/in-dia/mi.

16 14,900

18 17,500

20 20,100

22 27,025

Table 2: Pipeline Installation Costs

The total pipe cost = PMC + Installation + Wrapping & Coating, where

wrapping and coating are usually 5% of the PMC. The cost of the compressor station

labor and materials is found from7

The amount of fuel gas required to operate the compressor station depends on

the amount of horsepower the compressor is operating at. The following equation

depicts the relationship between fuel gas requirements and the operating horsepower at

the compressor station at three specific pipe diameters4.

Y = 0.0002x + 0.0006

The data points used to derive the linear relationship were obtained from Gas

Pipeline Hydraulics, by Menon. The equation allows the fuel requirement to be found

for a wide range of compressor sizes.

The total capital costs are then calculated from equipment, miscellaneous costs,

right of way, acquisitions, SCADA, and telecommunication systems, and other initial

costs. The total annual capitalized costs include capital and annual fuel costs.

The following derivation is for the Total Annual Costs, which is the annual cost

the project will cost at an interest rate of 8%. To find the annualized capital cost of the

fuel and the equipment, start with the present worth (P) for a stream of equal annual

cash flows (A) over the course of n years at an interest rate of i.

Equation 1

Multiply both sides by

Equation 2

Subtract Equation 1 from Equation 2.

Then solve for P and simplify.

Then solving for A yields the amount needed annually to recover the present worth of

the capital cost (P) of a project that is expected to last n years at an interest rate of i.

Next, the following equation is used to find the total annual cost per MCF4

Where Q = flow rate, MMSCFD assuming 350 operating days per year.

In this procedure, the optimum pipe diameter is identified by corresponding to the

curve with the lowest cost, which ideally should also have a flat profile over the range

of flow rates of interest. Figure 6 shows curves for a single, unbranched pipeline with a

feed at the inlet and a single consumer point at the outlet.

To exemplify the use of J-Curves we consider the design of a pipeline between a

supplier point and consumer point (single pipe case), as seen in Figure 5. In the

simulations, the compressor power was varied so that the pressure at the pipe outlet was

800 psig for the different pipe diameters and flow rates.

Figure 5: Simulation setup used to create J-curves in Figure 6.

Figure 6: J-Curves for the design of a single pipe segment.

For this situation, if the system was intended to have a flow rate of 300 MMSCFD, then

a pipe diameter of 16 inches would be the economic optimum with a Total Annual Cost

of $ 0.58 per MCF delivered.

$0.58$0.0

$1.0

$2.0

$3.0

$4.0

$5.0

50 100 150 200 250 300 350 400 450 500

To

tal A

nnual

Co

st p

er M

CF

Flow Rate (MMSCFD)

J-Curve

NPS = 18

NPS = 16

NPS = 20

NPS = 22

Limitations of the Existing Procedure

Several limitations to using J-Curves to select a pipe diameter and compressor

size can be identified.

1. The J-Curve is designed with a different compressor at each point along the

graph, so when a compressor is selected, the only accurate cost is at a single

point. A more accurate method is to select a compressor, and find the new costs

for several flow rates using a single compressor size. By selecting a compressor,

the capital cost of the compression station is fixed for all flow rates. However,

different flow rates require different powers at which the compressor operates

resulting in varied operating and fuel costs. In addition, the compressor operates

at different efficiencies when it is operated at different flow rates than that for

which it was designed.

2. The Panhandle equations have some errors. Simulation software can be a more

accurate and direct estimation of these pressure drops and compressor power

requirements. The Panhandle equations do not account for pipe roughness in

their pressure drop analysis4.

3. The range of flow rates used and the criterion of choosing the lowest curve

works well when the curves do not cross. Otherwise, if the curves cross, one

needs to pay more attention to the frequency at which the pipeline will operate

for each flow rate. Although this is not described in the open literature, we

assume that some consideration is given to this issue in practice.

4. The J-curve procedure seems straightforward for one pipe and one compressor.

It is unclear how to use it for a long pipeline with multiple compressors,

branched pipelines, gathering and distribution systems, etc. It seems that each

section is treated separately after the location of the compressors is decided

somehow, without looking at how one set of decisions for one section affects the

other. We assume that current practice does this but we are unaware of existing

procedures to decide the compressors locations.

Our Extensions of the Existing Procedure

To make fair comparisons with the J-Curve procedure, a few modifications have been

introduced.

1. The use of the Panhandle equation is substituted by computer simulations using

more accurate pressure drop correlations for the subsequent J-curves.

2. For each J-curve, one compressor is picked, such that the same compressor can

handle all flow rates. This is usually the compressor at the largest flow rate

3. Efficiency changes in the selected compressor due to operating at different flow

rates are taken into account.

Illustration

The modified J-Curves are illustrated for a one pipe network and compared with the

above presented literature conventional J-curve with changing compressor sizes. First a

constant compressor efficiency of 0.8 is assumed. The extended curves represent a

more realistic prediction of the cost of operating a pipeline by fixing the compression

station installed power. The curves are only as long as they are, because above the end

of the line is the stonewall point, and on the lower end is the surge point. Outside of

these bounds the compressor acts unpredictably and very inefficiently.

$0.2

$0.7

$1.2

$1.7

$2.2

30 130 230 330 430

To

tal A

nnual

Co

st p

er M

CF

Flow Rate (MMSCFD)

Total Cost per MCF NPS = 16

Range of flow rates and costs for a compressor designed at Q = 200

Original J-curve

Q = 300 Q = 400

Figure 7: J-Curves for the design of a single pipe segment considering fixed

compressor sizes.

While both graphs indicate the same optimum, the evidence is less convincing in this

extended curve. The total annual costs change significantly with flow rate, which is

markedly different from the conventional curve. However, neither this extended graph

nor the conventional J-curve considers the effects of flow rate on the compressor

efficiency. When compressors operate above or below their design capacity, their

efficiency decreases, which requires larger amounts of power for operation.

Figure 8 uses efficiency estimations based to take into account decreased

efficiencies of flow rates higher and lower than the compressor was designed to

process8. Figure 7 shows only the flow rates that the selected compressor can achieve,

providing a more realistic look at what is possible with the selected compressor.

Figure 8: J-Curves for the design of a single pipe segment for different compressors

sizes and considering efficiency changes.

In order to analyze the effects of the efficiency analysis, the graph below

compares the ‘Extended J-Curves’ for a pipe size of NPS 18 with and without

0.4

0.6

0.8

1

1.2

1.4

1.6

100 150 200 250 300 350 400 450 500 550

To

tal A

nnual

Co

st p

er M

CF

Flow Rate (MMSCFD)

Total Cost per MCF NPS = 16

Considering Efficiency as a function of flow rate at Q = 400

Fixed Efficiency of 80% at Q = 400(solid line)

efficiency. The TAC for each curve is consistently higher when considering efficiency.

This is expected, since the actual work required is greater than the ideal work.

We now extend the method to a two compressor case with one consumer point

half way. J-curves can be constructed, but now the intermediate pressure P3 is a

variable. Pro-II simulations were run at several diameters for varying flow rates, and P3

pressures.

Figure 9: Two segment pipeline with a consumer point in the middle.

In a complex network, Pro-II simulations paired with J-curve analysis allows each

segment to be optimized. However, when designing the network, the optimum arrived

at by picking the pressure parameter P3 based on one segment can lead to an incorrect

optimum (this case is outlined below the graphs). In this two-segment pipeline, three

pressures were chosen for analysis: P3 = 750, 800, and 850 psig. When these values

were used (using Pro-II) for four different pipe diameters, their resulting pressure drops

and compressor power requirements were used to construct the J-curves shown in

Figure 10.

Figure 10: J-Curves for the design of segment 1. a) P3=750psi; b) P3=800psi; c)

P3=850psi.

Optimizing Segment 1 at each pressure

0.3287

$0.00

$0.40

$0.80

$1.20

$1.60

100 150 200 250 300 350 400 450 500

TA

C p

er M

CF

Flow Rate (MMSCFD)

A) Segment 1P = 750

NPS = 16

NPS = 18

NPS = 20

NPS = 22

0.353

$0.00

$0.40

$0.80

$1.20

$1.60

100 150 200 250 300 350 400 450 500

TA

C p

er M

CF

Flow Rate (MMSCFD)

B) Segment 1P = 800

NPS = 16

NPS = 18

NPS = 20

NPS = 22

Considering each of the three pressures and each diameter for segment 1, the

optimum pressure was found to be 750 psig, and the best pipe diameter was 18 inches.

Figure 11: J-Curves for the design of segment 2. a) P3=750psi; b) P3=800psi; c)

P3=850psi

0.356

$0.34

$0.00

$0.40

$0.80

$1.20

$1.60

100 200 300 400 500

TA

C p

er M

CF

Flow Rate (MMSCFD)

C) Segment 1P = 850

NPS = 16NPS = 18NPS = 20NPS = 22

0.3063

0.3026

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

100 200 300 400 500

TA

C p

er M

CF

Flow Rate (MMSCFD)

A) Segment 2P = 750

NPS = 16

NPS = 18

NPS = 20

NPS = 22

.

Considering each of the three pressures and each diameter for segment 1, the

optimum pressure was found to be 850 psig, and the best pipe diameter was 18 inches.

0.293

0.2903

$0.00

$0.40

$0.80

$1.20

$1.60

100 150 200 250 300 350 400 450 500

TA

C p

er M

CF

Flow Rate (MMSCFD)

B) Segment 2P = 800 NPS = 16

NPS = 18

NPS = 20

NPS = 22

0.2801

0.2791

$0.00

$0.20

$0.40

$0.60

$0.80

$1.00

$1.20

$1.40

$1.60

100 150 200 250 300 350 400 450 500

TA

C p

er M

CF

Flow Rate (MMSCFD)

C) Segment 2P = 850

NPS = 16

NPS = 18

NPS = 20

NPS = 22

Figure 12: Combined J-Curves (segment 1 + segment 2). a) Overall optimum found by

optimizing segment 1 first; b) Overall optimum found by optimizing segment 2 first

By optimizing segment 1 first and selecting a pressure of 750 psig and a

diameter of 18 inches based on the optimum for segment 1, the optimum pipe diameter

for segment 2 with that pressure is 18 inches. The Total Annual Cost per MCF is $

0.631 for this configuration. But, by optimizing segment 2 first and selecting a pressure

of 850 psig and a pipe diameter of 18 inches based on the optimum for segment 2, then

the optimum diameter for segment 1 with that pressure is 18 inches. The TAC per MCF

for this arrangement would be $ 0.607. It then becomes apparent that in order to get the

optimum, all possibilities have to be examined in order to find the best optimum.

0.631

$0.30

$0.50

$0.70

$0.90

$1.10

$1.30

$1.50

$1.70

0 100 200 300 400 500 600

TA

C p

er M

CF

Flow Rate (MMSCFD)

A) Optimizing Segment 1P = 750

NPS = 18 Segment 1 & NPS = 18 Segment 2

0.616

$0.40

$0.60

$0.80

$1.00

$1.20

$1.40

$1.60

50 150 250 350 450 550

TA

C p

er M

CF

Flow Rate (MMSCFD)

Optimizing Segment 2P=850


Figure 13: J-Curves of the total design.

Figure 13 above shows the overall optimum which occurs with a pipe size of 18 inches

in both segments and an intermediate pressure of 850 psig. To find the overall

optimum, there are 12 more curves for the intermediate pressure of 850 psig to account

for all the diameter combinations. And then, there are 16 more curves for each of the

two other intermediate pressures to come to a total of 48 J-curves to analyze.

This illustrates the importance of analyzing the network as a whole in order to

determine the best combination of all parameters. For this simple two-pipe network, 18

total J-curves were required to analyze the system. For a larger network, it would be

impossible to know which segment should be optimized first in order to arrive at the

overall solution. As the number of segments grows, the number of simulations and J-

curves required grows exponentially.

These unnecessary complications are time-consuming as well as misleading.

For this simple two-pipe network, the following parameters influence the complexity:

1. 9 flow rates

2. 3 pressures

3. 4 diameters

0.6174

0.6164

$0.2

$0.6

$1.0

$1.4

$1.8

$2.2

100 150 200 250 300 350 400 450 500

TA

C p

er M

CF

Flow Rate (MMSCFD)

Overall Optimum NPS 18P=850





This requires 432 simulations in order to arrive at a J-curve which does not

provide accurate costs. A four-pipe segment would require 62,208 simulations which

could easily take weeks to run the simulations and analyze the output for this very

simple situation with no variable demand. Since most pipeline networks are far more

complicated than a four-pipe segment, using Pro-II simulations to formulate J-curves is

impractical and possibly inaccurate. The time required to perform this analysis

increases exponentially for an increasingly complex network. This method does not

even take into account variable demand during the project lifetime. The optimums are

chosen for a specific flow rate, yet rarely does demand remain constant. A more

inclusive method is required to evaluate even these simple pipeline networks accurately.

MATHEMATICAL PROGRAMMING MODEL

For the mathematical model, a correlation has to be used in the model to determine the

pressure drop in a given pipe. This correlation is shown below.

Where Q is the flow rate in M m3 per day

L is the length (km)

D is the pipe inner diameter (in)

P2 is the outlet pressure (kPa)

P1 is the inlet pressure (kPa)

ΔZ is the elevation difference (m)

A and B are determined by a regression analysis by graphing Q2L/D

5 (referred to as

parameter F) versus (P12-P2

2), and A will be the slope and B is the y intercept divided by

-ΔZ. Simulation software (Pro-II) was used to find (P12-P2

2) for a range of Q, L and D.

Figure 14: Deriving A and B values from Pro-II simulation data

The values for A and B were determined for different pipe diameters and for

each pipe diameter for no elevation change, and a positive and a negative elevation

change. So, for four pipe diameters, nine sets of A and B were found, which are shown

in Figure 15 below.

Figure 15: Pressure drops calcuated using A and B in the Empirical equation compared

to pressure drops predicted by Pro-II simulation data.

y = 67.826x - 2E+07R² = 0.9999

0

500

1000

1500

2000

2500

3000

0 10 20 30 40 50

Mill

ion

s

P22-P3

2Millions

Segment 2NPS = 28

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 50 100 150 200 250 300 350 400

Pre

ssure

Dro

p (

kP

a)

Flow Rate (MMSCFD)

Pro-II Pressure Drop vs. Empirical Pressure Drop

16; Pro-II 16; Analytical




There is a relatively small amount of error in the A and B factors for this flow

rate, all less than 5 % error. This leads to the idea that there may be ranges of (P22-P1

2)

in which certain values of A and B are valid.

The advantage to using this correlation for finding the pressure drop, is that once

the values for A and B are determined with sufficient accuracy, they may be used by the

mathematical modeling software along with the same equations above that go from

pressure drop to compressor work to compressor cost and pipeline cost to finally select

the optimum pipe diameter and compressor for each segment. Then, whichever

conditions are selected by the mathematical model can be used in simulation software to

find a more accurate pressure drop and a more accurate work required by each

compressor.

5. EXAMPLES

Before we use the model to solve larger problem, the two cases (single pipe and

two segments pipeline) analyzed by J-curves will be also solved by the proposed

mathematical model. Then two larger examples are presented: a linear network and a

branched network.

5.1 - Single Pipe Case

The mathematical model was run for a single pipe segment of 120 mi with a

flow rate of 200, 300 and 400 MMSCFD. The model predicted the following:

Mathematical Model Pressure Drop Error

Q

MMSCFD

dP Model

(kPa)

dP Pro-II

(kPa) % Error

200 4975 5055 1.62

300 7425 7326 1.34

400 9950 9845 1.05

Table 3

In order to determine the accuracy of the mathematical model, the conditions of the

optimal results predicted by the model were input into Pro-II simulation software. All

conditions and parameters were maintained to ensure comparability. The resulting

pressure drops predicted by simulations are compared in the table above, resulting in

minimal errors. This ensures the accuracy and reliability of results obtained through the

mathematical model.

5.2 – Two Segments Pipeline

The mathematical model was then used to optimize the two-pipe network

discussed previously. The model considered four pipe diameters and optimized the

network under the following conditions:

Non Linear Model – 2 Pipe Network

Pipe 1 Pipe 2

Pipe Diameter

(in) 22 22

Compressor

Work (hp) 10,740 0

Pressure Drop

(psi) 1,830 1,490

TAC Model $ 0.596

TAC J-Curves $ 0.616

Table 4

The model was able to predict the optimum pipe diameters for each segment, along with

the size of the required compressors and their locations. All consumer points were

considered for adding a compressor station, yet the model predicted the need for only

one compressor at the supply point.

5.3 – Linear Pipeline

Figure 16

The following problem is solved applying the different proposed models and the

solutions are presented and discussed: Consider one wants to find the most profitable

pipeline planning to delivery natural gas to nine consumers from two different supplier

points over a twenty years life time project.

The gas is composed by 91.9% of methane, 5% of ethane, 2% of propane, 1% of n-

butane and 0.1% of n-pentane.

Table 5 presents the distances between supplier and consumers, and Table 6

presents the distances between consumers.

Table 5 – Distances between suppliers and consumers.

Distance

(km) Cons 1 Cons 2 Cons 3 Cons 4 Cons 5 Cons 6 Cons 7 Cons 8 Cons 9

Supplier 1 115 143 323 550 609 613 630 638 650

Supplier 2 28 0 180 407 466 470 487 495 507

Table 6 – Distances between consumers.

Distance

(km) Cons 1 Cons 2 Cons 3 Cons 4 Cons 5 Cons 6 Cons 7 Cons 8 Cons 9

Cons 1 0 28 208 435 494 498 515 523 535

Cons 2 28 0 180 407 466 470 487 495 507

Cons 3 208 180 0 227 286 290 307 315 327

Cons 4 435 407 227 0 59 63 80 88 100

Cons 5 494 466 286 59 0 4 21 29 41

Cons 6 498 470 290 63 4 0 17 25 37

Cons 7 515 487 307 80 21 17 0 8 20

Cons 8 523 495 315 88 29 25 8 0 12

Cons 9 535 507 327 100 41 37 20 12 0

Table 7 – Elevation at the suppliers and consumers.

Station Elevation (m)

Supplier 1 42

Supplier 2 14.93

Consumer 1 7

Consumer 2 14.93

Consumer 3 60

Consumer 4 10

Consumer 5 120

Consumer 6 122

Consumer 7 235

Consumer 8 470

Consumer 9 890

Table 8 shows the conditions of the natural gas at the supplier stations, the

current price, annual price increase and penalty agreed by contract.

Table 8 – Suppliers data.

Station Pressure (kPa) Temperature (ºC) Price (US$/m^3) Penalty

(US$/m^3)

Supplier 1 1367 30 270 80

Supplier 2 1520 35 300 50

Table 9 – Consumers data.

Station Demand (m^3/day) Annual demand

increasing Price (US$/m^3)

Consumer 1 2,148,200 2% 440

Consumer 2 2,832,000 5% 400

Consumer 3 18,240 4% 390

Consumer 4 6,595,200 1% 420

Consumer 5 134,400 6% 445

Consumer 6 384,000 3% 460

Consumer 7 336,000 2% 420

Consumer 8 3,617,400 4% 410

Consumer 9 64,600 6% 450

Non Linear Model – 9 Pipe Network

Pipe 1 2 3 4 5 7 6 8 9

Pipe

Diameter

(in)

36 36 36 32 32 24 24 24 24

Compressor

Work (hp) 21,895 685 0 0 0 0 0 0 0

Table 10

The model created a network with expected results. The pipe diameters are highest

at the beginning where the flow is greatest, and there are only two compressors at the

first two points that should be built at the beginning of the project. The Total Annual

Cost for this network is 142 million dollars per year. This model takes less than two

minutes to run.

5.4 - Branched Pipeline Network

Figure 17

The conventional method of using J-curves to choose an optimum pipe diameter

is time-consuming and introduces error into the analysis. When using Pro-II, the

number of simulations is impractical for even four pipe segments. In most cases,

networks are arranged in a complex form, such as the network shown below. This

network consists of 8 pipes, three pipe diameters, and three pressures. To analyze a

complicated network like this, it would take 483

79 or 1.1 billion simulations, requiring

approximately ten years working non-stop.

Figure 18

The ramified network could account for increasing demand over time. The

mathematical model could more easily handle a network larger than this, with even

greater complexity.

Non Linear Model – Ramified Network

Pipe Pipe

S1-C1

Pipe

C1-C2

Pipe

C2-C3

Pipe

S2-S4

Pipe

S2-S5

Pipe

C5-C6

Pipe

C5-C7

Pipe Diameter

(in) 24 24 24 24 28 28 24

Compressor Work

(hp) 5,010 0 0 8,350 8,350 0 0

Pressure Drop

(psi) 65 4,190 4,255 180 30 100 10

Table 11

The model chose to put compressors at both supplier points at the beginning of

the project and to only use pipe sizes 24 and 28. Another important note is that the

model did not connect consumer points 1 and 4 with a pipe segment, indicating that

excluding a connection results in a more economically optimized network.

5.5 Linear Mathematical Model-Results

A linear mathematical model was also considered for pipeline network

optimization. This model relaxes the pressure parameters found in the non-linear

model, allowing the upper and lower bounds of the system to be defined. In order to

relax the pressures, the linear model discretizes the pressures into as many discrete units

as precision demands.

The linear model was used to optimize the single pipe network discussed

previously, with the results shown in Table 12.

Linear Mathematical Model – Single Pipe Segment

TAC

(millions)

NPV

(millions)

Time

(sec) Pipe S1-C1 NPS

Upper

bound 63.5 -192.5 4.547 10932 22

Lower

bound 47.6 -179.7 3.389 7922 22

Table 12

The upper and lower bounds predicted the range of possible solutions for the single pipe

network. Both limits required a compressor at the supply.

The linear mathematical model was then used to optimize the two-pipe segment

discussed previously. Table 13 shown below highlights the results.

Linear Mathematical Model – Two Pipe Segment

TAC

(millions)

NPV

( millions)

Time

(sec)

Compressor

Work

Pipe 1

Compressor

Work

Pipe 2

NPS

Pipe 1

NPS

Pipe 2

Upper

bound 87.98 -177.1 1219 15828 0 20 20

Lower

bound 71 -184.3 1613 9242 3240 22 20

Table 13

From Table 13, it is apparent that the linear mathematical model predicts the

requirement of two compressor stations, one at the supply and on at consumer point 1.

The upper bound predicts only a single compressor station at the supply point. As

expected, the upper bound predicted a higher Total Annual Cost per MCF.

The linear nine-pipe segment previously analyzed by the non-linear mathematical model

was then analyzed by the linear model, with the following results:

Linear Mathematical Model – Nine Pipe Segment

NPS

Pipe 1

NPS

Pipe 2

NPS

Pipe 3

NPS

Pipe 4

NPS

Pipe 5

NPS

Pipe 6

NPS

Pipe 7

NPS

Pipe 8

NPS

Pipe 9

Upper bound

pipe size 32 28 28 28 24 28 24 24 24

Lower

bound pipe

size

32 28 28 28 24 28 24 24 24

Upper bound

Compressor

Work

19421 9556 0 8755 0 0 0 0 0

Lower

bound

Compressor

Work

19421 9556 0 8755 0 0 0 0 0

Table 14

As shown in Table 14, the upper and lower bounds predicted the same optimized pipe

diameters. The range of Total Annual Costs per MCF was found to be $376 - $476

million. Both limits predict the same required compressor station sizes and locations.

Finally, the ramified network previously discussed was optimized by the linear

mathematical model, with the results shown in Table 15 below.

Linear Mathematical Model – Ramified Network

Mathematical

Model

Pipe

S1-C1

Pipe

C1-C2

Pipe

C2-C3

Pipe

C1-C4

Pipe

S2-C4

Pipe

S2-C5

Pipe

C5-C6

Pipe

C5-C7

Upper bound

pipe size 24 24 24 None 24 36 24 24

Lower bound

pipe size 24 24 24 None 24 36 24 24

Upper bound

Compressor

Work

7072.3 1690 13.85 0 12432 12432 0 0

Lower bound

Compressor

Work

4705.3 1690 13.85 0 8017 8017 0 0

Table 15

From the linear mathematical model, the range of Total Annual Costs per MCF was

predicted to be $96 to $130 million. Both the upper and lower limits analysis predicted

the same compressor station locations, but with varied compressor station sizes. The

linear mathematical model predicted the exclusion of a pipe segment from consumer

point 1 to consumer point 4, which is consistent with the prediction of the non-linear

model.

The accuracy of the linear mathematical model’s upper and lower bound

predictions has not been verified. However, the non-linear mathematical model

predicted network optimum conditions to a high degree of accuracy. Although, it

should be noted that the mathematical models need to be used with great care because it

is only as accurate as the input that the model is given.

6. EXTENSIONS

Contracts:

o Suppliers – It is being assumed that price and purchased amount of

gas is previously agreed. Also, a penalty is applied when the agreed

amount of gas is not bought. This agreed amount is an important

parameter and will be handled as a decision variable.

o Consumers – The contract with consumers also define a price, and

penalty due to not delivery the required demand. Demand will be

handled as input parameter. In this contract, the penalty per cubic

meter not delivered is 10% of the agreed price.

In both case, price readjustments (from prices forecast) were not directly

considered since both sides will have the same inflation influence. However,

the value of the money still be considered when a discount factor is applied.

7. CONCLUSIONS

In conclusion, J-Curves can be very time consuming and in some instances

misleading. To accurately examine all the parameters together requires a very large

amount of time and simulations that will only be applicable for that one situation. It is

worthwhile to investigate a model that could be adaptable to many other designs. The

mathematical model analyzed in this paper shows promise and could become a very

useful tool for pipeline network design.

Nomenclature

Parameters

DF – Discount factor

VCC – Coefficient related with the variable part of compressors capital cost

FCC – Coefficient related with the fixed part of compressors capital cost

PC – Parameter related with the cost of pipe of different diameters

SPrice – Prices charged by each supplier

CPrice – Prices charged for each consumer

OPC – Compressors operating cost

OPH – Annual operating hours

LCC – Distance between two consumers

LSC – Distance between supplier and consumer

ID – Inside diameters options

A – Pre pressure drop parameter in the empirical equation

B – Pre elevation parameter in the empirical equation

ZC – Elevation of each consumer

ZS – Elevation of each supplier

UB – Flowrate upper bound at each time (defined by the summation of demands each

year)

Demand – Demand required by each consumer in each year

CPenal – Penalty applied when demand is not delivered

SPenal – Penalty applied when the agreed amount is not bought

k – Cp/Cv

ST – Suction temperature at each supplier

SP – Suction pressure at each supplier

Z – Compressibility factor, dimensionless

η – Compressor efficiency

Q – volumetric flow rate (SCFD)

E – pipeline efficiency

Pb – base pressure (psia)

Tb – base temperature (°R)

P1 – upstream pressure (psia)

P2 – downstream pressure (psia)

Tf the average gas flow temperature (°R)

Le the equivalent length of pipe segment (miles)

G – gas gravity

e = base of natural logarithms (e=2.718…)

s = elevation adjustment parameter, dimensionless

H1 is the upstream elevation (m)

H2 is the downstream elevation (m)

Tamb

– Ambient temperature

ρsta

– Density at standard conditions

C – Empirical constant of erosion velocity equation

α – Angular coefficient of the empirical equation for density

β – Linear coefficient of the empirical equation for density

Variables

NPV – Net present value

Revenue - Annual revenue

FCI – Annual fixed capital of investment

Oper – Operating cost at each year

Penal – Penalties paid at each year

WS – Work done by the compressor at each supplier at each year

WC – Work done by the compressor at each consumer station at each year

DP – Discharge pressure of each compressor at the supplier station

Caps – Compressor capacity at each supplier point

Capc – Compressor capacity at each consumer point

Agreed – Amount of gas agreed to purchase from each supplier

XDCC – Binary variable related to the diameter size and installation time of a pipeline

between two consumers points

XDSC – Binary variable related to the diameter size installation time of a pipeline

between a supplier point and a consumer point

XCC – Binary variable related to time when a compressor is installed at the consumer

XCS – Binary variable related to time when a compressor is installed at the supplier

YDCC – Binary variable related to the operation conditions of the pipeline between two

consumers points at each time

YDSC – Binary variable related to the operation conditions of the pipeline between a

supplier and a consumer at each time

YCC – Binary variable related to the operation conditions of the compressor installed at

the consumer

YCC – Binary variable related to the operation conditions of the compressor installed at

the supplier

XPC – Binary variable related the pressure at each consumer at each time

QC – Flowrate delivered at each consumer point

QSC – Flowrate between a supplier point and a consumer point

QCC – Flowrate between a two consumer points

Parameters

PXCC – Binary parameter related to the allowed existence of a compressor at consumer

points

PXSC – Binary parameter related to the allowed existence of a compressor at supplier

points

XPCC – Binary parameter related to the allowed existence of a pipeline between two

consumers points

XPSC – Binary parameter related to the allowed existence of a pipeline between a

supplier point and a consumer point

References

Edgar, Himmelblau and Bickel. Optimal design of Gas transmission Networks, SPE

Journal, 1978Menon, Shashi.

Mohitpour, Golshan and Murray. Pipeline Design and Construction – A Practical

Approach. 2003

Peters, Timmerhaus, and West. Plant Design and Economics for Chemical Engineers.

©2003 by McGraw-Hill Companies Inc.

Fourth Edition, ©2001 by John Wiley & Sons, Inc.

J.Vincent-Genod. Fundamentals of Pipeline Engineering. ©1984 by Technip.

1Energy Information Administration www.eia.doe.gov

2Lyons, Plisga. Standard Handbook of Petroleum & Natural Gas Engineering (2nd

Edition). 2005 by Elsevier

3McAllister. Pipeline Rules of Thumb Handbook - A Manual of Quick, Accurate

Solutions to Everyday Pipeline Engineering Problems (6th Edition). 2005 by Elsevier

4Gas Pipeline Hydraulics, E. Shashi Menon, © 2005 Taylor & Francis Group, LLC

5http://www.uniongas.com/aboutus/aboutng/composition.asp

6Welty, Wicks, Wilson, Rorrer. Fundamentals of Momentum, Heat and Mass Transfer,

7Scribner, Faria, and Bagajewicz. Internal Report Pipeline Cost Estimation. University

of Oklahoma August 2007.

8Vincent-Genod. Fundamentals of Pipeline Engineering. Gulf Pub Co, 1984.

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